CPrule: Constrained proportional rule

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Constrained proportional rule

Description

CPrule calculates the contribution vector selected by the CP rule.

Usage

CPrule(c)

Arguments

c	A numeric cost vector.

Details

For each c\in C^N, let c_0=0 and (q_0,q_1,\dots,q_s)\in\mathbb{N}^{s+1}, with 0=q_0<q_1<\dots<q_s=n, defined recursively, for j\geq 0, by

q_{j+1}=\text{max}\Bigg\{q\in N_{+}^{q_j}:\dfrac{c_q-c_{q_j}}{c_{q_j+1}+\dots+c_q}=\text{min}\bigg\{\dfrac{c_r-c_{q_j}}{c_{q_j+1}+\dots+c_r}:r\in N_{+}^{q_j}\bigg\}\Bigg\}.

Then, for each j\in\{0,\dots,s-1\} and each i\in Q_j=\{q_j+1,\dots,q_{j+1}\},

\text{CP}_i(c)=\dfrac{c_i}{c(Q_j)}(c_{q_{j+1}}-c_{q_j}).

With this rule, calculating each agent's contribution is not always straightforward. When a coalition of agents violates the NS constraint, it becomes necessary to proceed in two or more steps.

The core idea of this rule is proportionality, aiming to ensure that agents' contributions are as close as possible to being proportional to the cost parameters, while respecting these constraints.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

References

Bernárdez Ferradás, A., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2025). Airport problems with cloned agents. [Preprint manuscript].

Thomson, W. (2024). Cost allocation and airport problems. Mathematical Social Sciences, 31(C), 17–31.

See Also

basicrule, weightedrule, clonesrule, hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
CPrule(c)

AirportProblems documentation built on June 8, 2025, 10:49 a.m.